us look at each constructor in turn. We use a nameless representation for variables — they are de Bruijn indexed. Variables are represented by a proof of their membership in the context, HasType i G T, Var : HasType i G t -> Expr G t So, in an expression \x,\y. x y, the variable x would have a de Bruijn index of 1, represented as Pop Stop, and y 0, represented as Stop. We find these by counting the language, any variable is translated to the Var constructor, using Pop and Stop to construct the de Bruijn index (i.e., to count how many bindings since the variable itself was bound); and any lambda is translated

us look at each constructor in turn. We use a nameless representation for variables — they are de Bruijn indexed. Variables are represented by a proof of their membership in the context, HasType i G T, Var : HasType i G t -> Expr G t So, in an expression \x,\y. x y, the variable x would have a de Bruijn index of 1, represented as Pop Stop, and y 0, represented as Stop. We find these by counting the language, any variable is translated to the Var constructor, using Pop and Stop to construct the de Bruijn index (i.e., to count how many bindings since the variable itself was bound); and any lambda is translated

us look at each constructor in turn. We use a nameless representation for variables — they are de Bruijn indexed. Variables are represented by a proof of their membership in the context, HasType i G T, Var : HasType i G t -> Expr G t So, in an expression \x,\y. x y, the variable x would have a de Bruijn index of 1, represented as Pop Stop, and y 0, represented as Stop. We find these by counting the language, any variable is translated to the Var constructor, using Pop and Stop to construct the de Bruijn index (i.e., to count how many bindings since the variable itself was bound); and any lambda is translated

AGDAPATTERN and AGDACLAUSE built-ins respectively. Types are simply terms. Terms and patterns use de Bruijn indices to represent variables. data Term : Set data Sort : Set data Pattern : Set data Clause : Pattern proj : (f : Name) → Pattern absurd : (x : Nat) → Pattern -- Absurd patterns have de Bruijn indices data Clause where clause : (tel : Telescope) (ps : List (Arg Pattern)) (t : correct-by-construction type checker for simply typed λ-calculus. First we define the raw terms, using de Bruijn indices for variables and explicit type annotations on the lambda: infixr 6 _=>_ data Type : Set

AGDAPATTERN and AGDACLAUSE built-ins respectively. Types are simply terms. Terms and patterns use de Bruijn indices to represent variables. data Term : Set data Sort : Set data Pattern : Set data Clause : Pattern proj : (f : Name) → Pattern absurd : (x : Nat) → Pattern -- Absurd patterns have de Bruijn indices data Clause where clause : (tel : Telescope) (ps : List (Arg Pattern)) (t : correct-by-construction type checker for simply typed λ-calculus. First we define the raw terms, using de Bruijn indices for variables and explicit type annotations on the lambda: infixr 6 _=>_ data Type : Set

AGDAPATTERN and AGDACLAUSE built-ins respectively. Types are simply terms. Terms and patterns use de Bruijn indices to represent variables. data Term : Set data Sort : Set data Pattern : Set data Clause : Pattern proj : (f : Name) → Pattern absurd : (x : Nat) → Pattern -- Absurd patterns have de Bruijn indices data Clause where clause : (tel : Telescope) (ps : List (Arg Pattern)) (t : correct-by-construction type checker for simply typed λ-calculus. First we define the raw terms, using de Bruijn indices for variables and explicit type annotations on the lambda: infixr 6 _=>_ data Type : Set

languages (DSLs) • Software verification • Formalized Mathematics • Dependent Type Theory • de Bruijn’s principle: small trusted kernel, external proof/type checkers Programming Language Resources

languages (DSLs) • Software verification • Formalized Mathematics • Dependent Type Theory • de Bruijn’s principle: small trusted kernel, external proof/type checkers Programming Language Lean Timeline

AGDAPATTERN and AGDACLAUSE built-ins respectively. Types are simply terms. Terms and patterns use de Bruijn 138 Chapter 3. Language Reference Agda User Manual, Release 2.6.2 indices to represent variables Literal) → Pattern proj : (f : Name) → Pattern absurd : (x : Nat) → Pattern -- Absurd patterns have de Bruijn indices data Clause where clause : (tel : Telescope) (ps : List (Arg Pattern)) (t : Term) → Clause correct-by-construction type checker for simply typed ?-calculus. First we define the raw terms, using de Bruijn indices for variables and explicit type annotations on the lambda: 3.38. Syntactic Sugar 161 Agda

AGDAPATTERN and AGDACLAUSE built-ins respectively. Types are simply terms. Terms and patterns use de Bruijn indices to represent variables. data Term : Set data Sort : Set data Pattern : Set data Clause : Literal) → Pattern proj : (f : Name) → Pattern absurd : (x : Nat) → Pattern -- Absurd patterns have de Bruijn indices data Clause where clause : (tel : Telescope) (ps : List (Arg Pattern)) (t : Term) → Clause correct-by-construction type checker for simply typed ?-calculus. First we define the raw terms, using de Bruijn indices for variables and explicit type annotations on the lambda: infixr 6 _=>_ data Type : Set